and Cosmology

7.2 Gravitational Instability
gravitational instability leads to an increase of
density fluctuations over the course of time. The
evolution of structure in the Universe is described
by the model of gravitational instability.
The evolution of structure in the Universe can be
understood in the framework of this model. In this chapter
we will describe structure formation quantitatively.
This includes the analysis of the evolution of density
perturbations over time, as well as a statistical description
of such density fluctuations. We will then see that
the evolution of inhomogeneities is directly observable,
and that the Universe was less inhomogeneous at high
redshift than it is today. Since the evolution of perturbations
depends on the cosmological model, we need
to examine whether this evolution can be used to obtain
an estimate of cosmological parameters. In Chap. 8, we
will give an affirmative answer to this question. Finally,
we will briefly discuss the origin of density fluctuations.
7.2.2 Linear Perturbation Theory
We first will examine the growth of density perturbations.
For this discussion, we will concentrate on
length-scales that are substantially smaller than the
Hubble radius. On these scales, structure growth can
be described in the framework of the Newtonian theory
of gravity. The effects of spacetime curvature and thus
of General Relativity need to be accounted for only for
density perturbations on length-scales comparable to, or
larger than the Hubble radius. In addition, we assume
for simplicity that the matter in the Universe consists
only of dust (i.e., pressure-free matter), with density
ρ(r, t). The dust will be described in the fluid approximation,
where the velocity field of this fluid shall be
denoted by v(r, t). 1
1 Strictly speaking, the cosmic dust cannot be described as a fluid
because the matter is assumed to be collisionless. This means that no
interactions occur between the particles, except for gravitation. Two
flows of such dust can thus penetrate each other. This situation can
be compared to that of a fluid whose molecules are interacting by
collisions. Through these collisions, the velocity distribution of the
molecules will, at each position, assume an approximate Maxwell
distribution, with a well-defined average velocity that corresponds to
the flow velocity at this point. Such an unambiguous velocity does not
exist for dust in general. However, at early times, when deviations from
the Hubble flow are still very small, no multiple flows are expected,
so that in this case, the velocity field is unambiguously defined.
Equations of Motion. The behavior of this fluid is
described by the continuity equation
∂ρ
+∇·(ρ v) = 0 , (7.2)
∂t
which expresses the fact that matter is conserved: the
density decreases if the fluid has a diverging velocity
field (thus, if particles are moving away from each
other). In contrast, a converging velocity field will
lead to an increase in density. Furthermore, the Euler
equation applies,
∂v
P
+ (v ·∇) v =−∇ −∇Φ, (7.3)
∂t ρ
which describes the conservation of momentum and
the behavior of the fluid under the influence of forces.
The left-hand side of (7.3) is the time derivative of the
velocity as would be measured by an observer moving
with the flow, because ∂v/∂t is the derivative at a fixed
point in space, whereas the total left-hand side of (7.3)
is the time derivative of the velocity measured along the
flow lines. The latter is affected by the pressure gradient
and the gravitational field Φ, the latter satisfying the
Poisson equation
∇ 2 Φ = 4πGρ. (7.4)
Since we are only considering dust, the pressure vanishes,
P ≡ 0. These three equations for the description
of a self-gravitating fluid can in general not be solved
analytically. However, we will show that a special,
cosmologically relevant exact solution can be found,
and that by linearization of the system of equations
approximate solutions can be constructed for |δ|≪1.
Hubble Expansion. The special exact solution is the
flow that we have already encountered in Chap. 4: the
homogeneous expanding cosmos. By substituting into
the above equations it is immediately shown that
v(r, t) = H(t)r
is a solution of the equations if ρ is homogeneous and
satisfies (4.11), and if the Friedmann equation (4.13) for
the scale factor applies.
As long as the density contrast |δ|≪1, the deviations
of the velocity field from the Hubble expansion will be
small. We expect that in this case, physically relevant
solution of the above equations are those which deviate
only slightly from the homogeneous case.
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